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Sun, 17 Jun 2018

Bounded does not imply totally bounded

I somehow managed to miss the notion of totally bounded when I was learning topology, and it popped up on stack exchange recently. It is a stronger version of boundedness for metric spaces: a space !!M!! is totally bounded if, for any chosen !!\epsilon!!, !!M!! can be covered by a finite family of balls of radius !!\epsilon!!.

This is a strictly stronger property than ordinary boundedness, so the question immediately comes up: what is an example of a space that is bounded but not totally bounded. Many examples are well-known. For example, the infinite-dimensional unit ball is bounded but not totally bounded. But I didn't think of this right away.

Instead I thought of the following rather odd example: Let !!S!! be the closed unit disc and assign each point a polar coordinate !!\langle r,\theta\rangle!! as usual. Now consider the following metric:

$$ d(\langle r_1, \theta_1\rangle, \langle r_2, \theta_2\rangle) = \begin{cases} r_1, & \qquad \text{ if $r_2 = 0$} \\ \lvert r_1 - r_2 \rvert, & \qquad\text{ if $\theta_1 = \theta_2$} \\ r_1 + r_2 & \qquad\text{ otherwise} \\ \end{cases} $$

The idea is this: you can travel between points only along the radii of the disc. To get from !!p_1!! to !!p_2!! that are on different radii, you must go through the origin:

A circle, with the
center marked.  A shortest-distance path is drawn in blue between two
blue points on the same radius, and in red between two red points on different
radii.  The blue path goes straight from one blue point to the other.
The red path goes from one point straight to the origin,
then straight to the other point.

Now clearly when !!\epsilon < \frac12!!, the !!\epsilon!!-ball that covers each point point !!\left\langle 1, \theta\right\rangle!! lies entirely within one of the radii, and so an uncountable number of such balls are required to cover the disc.

It seems like this example could be useful in other circumstances too. Does it have a name?

[ Addendum 2018-07-18: Several Gentle Readers have informed me that this metric has not just one name, but several. ]

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